Chu . o . ng 3 Khˆong gian vector 3.1 Kha´i niˆe . m vˆe ` khˆong gian vector 3.1.1 D - i . nh nghı ˜ a khˆong gian vector D - i . nh nghı ˜ a 3.1. Cho mˆo . t tˆa . p ho . . p E kha´c rˆo ˜ ng va` mˆo . t tru . `o . ng sˆo ´ T cu`ng v´o . i hai phe´p toa´n: - Phe´p cˆo . ng: E × E −→ E (x, y) −→ x + y. - Phe´p nhˆan ngoa`i T × E −→ E (λ, x) −→λx. E cu`ng v´o . i hai phe´p toa´n trˆen lˆa . p tha`nh mˆo . t khˆong gian vector trˆen K, hay K- khˆong gian vector nˆe ´ u 8 tiˆen d¯ˆe ` sau d¯ˆay d¯u . o . . c thu . . c hiˆe . n: (1) (x + y) + z = x + (y + z); ∀x, y, z ∈ E; (2) ∃0 E ∈ E sao cho: x + 0 E = 0 E + x = x; ∀x ∈ E; (3) ∀x ∈ E, ∃ − x ∈ E sao cho: x + (−x) = (−x) + x = 0 E ; (4) x + y = y + x; ∀x, y ∈ E; (5) λ(x + y) = λx + λy; ∀x, y ∈ E; ∀λ ∈ K; (6) (λ + µ)x = λx + µx; ∀x ∈ E; ∀λ, µ ∈ K; (7) (λµ)x = λ(µx); ∀x ∈ E; ∀λ, µ ∈ K; 47 48 3. Khˆong gian vector (8) 1x = x, ∀x ∈ K. Mˆo ˜ i phˆa ` n tu . ’ cu ’ a E d¯u . o . . c go . i la` mˆo . t vector, mˆo ˜ i sˆo ´ thuˆo . c K go . i la` mˆo . t vˆo hu . ´o . ng. 3.1.2 V`ai v´ı du . . a. Tˆa . p ho . . p V = Mat m×n (K) ca´c ma trˆa . n cˆa ´ p m × n trˆen tru . `o . ng K cu`ng v´o . i phe´p toa´n cˆo . ng hai ma trˆa . n, nhˆan mˆo . t sˆo ´ cu ’ a tru . `o . ng K v´o . i mˆo . t ma trˆa . n la` mˆo . t K- khˆong gian vector. Vector −→ 0 la` ma trˆa . n O, vector d¯ˆo ´ i −A la` ma trˆa . n d¯ˆo ´ i cu ’ a ma trˆa . n A. b. Cho V la` tˆa . p ho . . p ca´c vector hı`nh ho . c v´o . i vector −→ 0 la` vector co´ mod¯un b˘a ` ng 0 va` co´ hu . ´o . ng tu`y y´, ta xa´c d¯i . nh phe´p cˆo . ng va` phe´p nhˆan ngoa`i trˆen V nhu . sau: Phe´p cˆo . ng: V × V −→ V ( −→ x , −→ y ) −→ −→ x + −→ y −→ x + −→ y d¯u . o . . c xa´c d¯i . nh theo quy t˘a ´ c hı`nh bı`nh ha`nh Vector d¯ˆo ´ i − −→ x la` vector cu`ng phu . o . ng v´o . i vector −→ x , co´ d¯ˆo . da`i b˘a ` ng d¯ˆo . da`i vector −→ x va` ngu . o . . c hu . ´o . ng v´o . i vector −→ x . Phe´p nhˆan ngoa`i v´o . i mˆo . t sˆo ´ : v´o . i α ∈ R, −→ x ∈ V , α −→ x la` mˆo . t vector cu`ng phu . o . ng v´o . i −→ x , co´ d¯ˆo . da`i b˘a ` ng tı´ch cu ’ a |α| v´o . i d¯ˆo . da`i cu ’ a −→ x va` co´ hu . ´o . ng cu`ng hu . ´o . ng v´o . i −→ x nˆe ´ u α > 0, ngu . o . . c hu . ´o . ng v´o . i −→ x nˆe ´ u α < 0. Dˆe ˜ thˆa ´ y r˘a ` ng tˆa . p V cu`ng v´o . i hai phe´p toa´n trˆen thoa ’ ma ˜ n 8 tiˆen d¯ˆe ` cu ’ a d¯i . nh nghı ˜ a khˆong gian vector. Vˆa . y V la` mˆo . t khˆong gian vector trˆen R. c. Cho tru . `o . ng K, v´o . i n ≥ 1, xe´t tı´ch D - ˆeca´c: K n = {(x 1 , x 2 , ., x n )/x i ∈ K, i = 1, 2, ., n} cu`ng hai phe´p toa´n: (x 1 , x 2 , ., x n ) + (y 1 , y 2 , ., y n ) = (x 1 + y 1 , x 2 + y 2 , ., x n + y n ) k(x 1 , x 2 , ., x n ) = (kx 1 , kx 2 , ., kx n ), k ∈ K. Dˆe ˜ thˆa ´ y K n cu`ng hai phe´p toa´n trˆen la` mˆo . t K− khˆong gian vector. Vector O = (0, 0, ., 0), vector d¯ˆo ´ i cu ’ a x = (x 1 , x 2 , ., x n ) la` −x = (−x 1 , −x 2 , ., −x n ). D - ˘a . c biˆe . t: Khi n = 1 thı` ba ’ n thˆan K cu ˜ ng la` mˆo . t K− khˆong gian vector. d. Tˆa . p ho . . p ca´c sˆo ´ thu . . c R v´o . i phe´p cˆo . ng sˆo ´ thu . . c va` phe´p nhˆan sˆo ´ thu . . c v´o . i sˆo ´ h˜u . u ty ’ la` mˆo . t Q− khˆong gian vector. e. Tˆa . p K[x] ca´c d¯a th´u . c mˆo . t biˆe ´ n hˆe . sˆo ´ trˆen K v´o . i phe´p cˆo . ng d¯a th´u . c va` phe´p nhˆan mˆo . t phˆa ` n tu . ’ thuˆo . c tru . `o . ng K v´o . i mˆo . t d¯a th´u . c la` mˆo . t K− khˆong gian vector. Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 3.1. Kha´ i niˆe . m vˆe ` khˆong gian vector 49 3.1.3 Mˆo . t sˆo ´ tı´nh chˆa ´ t d¯o . n gia ’ n cu ’ a khˆong gian vector. Cho V la` mˆo . t K− khˆong gian vector tu`y y´. Khi d¯o´, ta luˆon co´: Tı´nh chˆa ´ t 3.1 (Tı´nh duy nhˆa ´ t cu ’ a phˆa ` n tu . ’ khˆong.). Chı ’ co´ duy nhˆa ´ t mˆo . t vector 0 ∈ V sao cho ∀x ∈ V : x + 0 = 0 + x = x. Thˆa . t vˆa . y, nˆe ´ u θ cu ˜ ng la` mˆo . t vector khˆong cu ’ a V thı`: θ = θ + 0 = 0. Tı´nh chˆa ´ t 3.2 (Tı´nh duy nhˆa ´ t cu ’ a phˆa ` n tu . ’ d¯ˆo ´ i.). V´o . i mˆo ˜ i x ∈ V , tˆo ` n ta . i duy nhˆa ´ t phˆa ` n tu . ’ d¯ˆo ´ i cu ’ a x la` −x sao cho: x + (−x) = 0. Thˆa . t vˆa . y, nˆe ´ u x  cu ˜ ng la` mˆo . t vector d¯ˆo ´ i cu ’ a x thı` : −x = −x + 0 = −x + (x + x  ) = (−x + x) + x  = 0 + x  = x  . Tı´nh chˆa ´ t 3.3. Luˆa . t gia ’ n u . ´o . c co´ hiˆe . u lu . . c trong V , t´u . c la`: +) (x + z = y + z) ⇒ (x = y), ∀x, y, z ∈ V ; +) (z + x = z + y) ⇒ (x = y), ∀x, y, z ∈ V. Thˆa . t vˆa . y, (x + z = y + z) ⇒ [(x + z) + (−z) = (y + z) + (−z)] ⇒ [x + (z − z) = y + (z − z)] ⇒ (x + 0 = y + 0) ⇒ (x = y). Tu . o . ng tu . . cho phˆa ` n co`n la . i. Tı´nh chˆa ´ t 3.4. ∀x, y, z ∈ V, (x + y = z) ⇔ (x = z − y). Thˆa . t vˆa . y, (x + y = z) ⇔ [(x + y) + (−y) = z + (−y)] ⇔ [x + (y − y) = z − y] ⇔ (x + 0 = z − y) ⇔ (x = z − y). Tı´nh chˆa ´ t 3.5. ∀λ ∈ K, ∀x ∈ V, λx = 0 ⇔  λ = 0 ∈ K x = 0 ∈ V Ch´u . ng minh. (⇐) λ0 = λ(0 + 0) = λ0 + λ0 ⇒ λ0 = 0 (theo luˆa . t gia ’ n u . ´o . c); 0x = (0 + 0)x = 0x + 0x ⇒ 0x = 0 (theo luˆa . t gia ’ n u . ´o . c). (⇒) Gia ’ su . ’ λx = 0 va` λ = 0. Khi d¯o´ ∃λ −1 ∈ K va` ta co´: x = 1x = (λ −1 λ)x = λ −1 (λx) = λ −1 0 = 0, t´u . c la` x = 0 ∈ V . Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 50 3. Khˆong gian vector Tı´nh chˆa ´ t 3.6. ∀λ ∈ K, ∀x ∈ V, −(λx) = (−λ)x = λ(−x). Thˆa . t vˆa . y, λx + (−λ)x = [λ + (−λ)]x = 0x = 0 = λx + [−(λx)] ⇒ (−λ)x = −(λx); λx + λ(−x) = λ[x + (−x)] = λ0 = 0 = λx + [−(λx)] ⇒ λ(−x) = −(λx) Vˆa . y: −(λx) = (−λ)x = λ(−x). 3.2 Khˆong gian vector con. D - i . nh nghı ˜ a 3.2. Mˆo . t tˆa . p ho . . p con W = ∅ cu ’ a K− khˆong gian vector V d¯u . o . . c go . i la` khˆong gian vector con cu ’ a V nˆe ´ u W ˆo ’ n d¯i . nh d¯ˆo ´ i v´o . i phe´p toa´n cˆo . ng va` phe´p nhˆan ngoa`i trˆen V . T´u . c la`, x + y ∈ W va` λx ∈ W v´o . i mo . i x, y ∈ W, mo . i λ ∈ K. D - u . o . ng nhiˆen khi W la` mˆo . t khˆong gian vector con cu ’ a V thı` W cu ˜ ng la` mˆo . t khˆong gian vector trˆen tru . `o . ng K. Vı´ du . . (1) K− Khˆong gian vector V la` mˆo . t khˆong gian con cu ’ a chı´nh no´ va` d¯u . o . . c go . i la` khˆong gian con khˆong thu . . c su . . . Tˆa . p ho . . p {0 V } chı ’ gˆo ` m mˆo . t vector khˆong cu ˜ ng la` mˆo . t khˆong gian vector con cu ’ a V va` d¯u . o . . c go . i la` khˆong gian con tˆa ` m thu . `o . ng cu ’ a V . Ta go . i khˆong gian con thu . . c su . . cu ’ a V la` mˆo . t khˆong gian con kha´c {0 V } va` kha´c V . (2) Nˆe ´ u coi C la` mˆo . t R− khˆong gian vector thı` R ⊂ C la` mˆo . t khˆong gian vector con cu ’ a C. Nˆe ´ u coi C la` mˆo . t C− khˆong gian vector thı` R khˆong la` mˆo . t khˆong gian vector con cu ’ a C vı` R khˆong ˆo ’ d¯i . nh v´o . i phe´p nhˆan v´o . i mˆo . t sˆo ´ ph´u . c. (3) Tˆa . p W = {a 0 + a 1 x + a 2 x x + · · · + a n x n |a i ∈ K} trong d¯o´ n la` mˆo . t sˆo ´ tu . . nhiˆen cho tru . ´o . c, la` mˆo . t khˆong gian vector con cu ’ a K− khˆong gian vector K[x]. D - i . nh ly´ 3.1. Cho W la` mˆo . t tˆa . p con kha´c rˆo ˜ ng cu ’ a K− khˆong gian vector V . Khi d¯o´ W la` mˆo . t khˆong gian vector con cu ’ a V khi va` chı ’ khi λx + µy ∈ W, ∀x, y ∈ W, ∀λ, µ ∈ K. Ch´u . ng minh. (⇒) Gia ’ su . ’ W la` khˆong gian con cu ’ a V . Khi d¯o´, ∀x, y ∈ W, ∀λ, µ ∈ K do λx, µy ∈ W nˆen λx + µy ∈ W . (⇐) Cho . n λ = µ = 1 thı` ∀x, y ∈ W , ta d¯ˆe ` u co´ x + y ∈ W ; Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 3.3. Su . . phu . thuˆo . c tuyˆe ´ n t´ınh v`a d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. 51 Cho . n λ = 1, µ = 0 thı` ∀x ∈ W, y = x, ta d¯ˆe ` u co´ λx + 0x = λx ∈ W. Do d¯o´ W la` mˆo . t khˆong gian vector con cu ’ a V . 3.3 Su . . phu . thuˆo . c tuyˆe ´ n t´ınh v`a d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. 3.3.1 Tˆo ’ ho . . p tuyˆe ´ n tı´nh va` biˆe ’ u thi . tuyˆe ´ n tı´nh. D - i . nh nghı ˜ a 3.3. Cho x 1 , x 2 , ., x n la` n vector (n ≥ 1) cu ’ a K− khˆong gian vector V va` λ 1 , λ 2 , ., λ n la` n vˆo hu . ´o . ng trong K. Vector x = λ 1 x 1 + λ 2 x 2 + · · · + λ n x n = n  i=1 λ i x i d¯u . o . . c go . i la` tˆo ’ ho . . p tuyˆe ´ n tı´nh cu ’ a hˆe . vector (x 1 , x 2 , ., x n ) = (x i ) i=1,n v´o . i ho . hˆe . sˆo ´ (λ 1 , λ 2 , ., λ n ) = (λ i ) i=1,n . Khi vector x la` mˆo . t tˆo ’ ho . . p tuyˆe ´ n tı´nh cu ’ a hˆe . (x i ) i=1,n thı` ta ba ’ o x biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . (x i ) i=1,n . Vı´ du . . Cho −→ x 1 = (1, −2), −→ x 2 = (3, 1), −→ x = (5, −3) ∈ R 2 . Ta co´ 2 −→ x 1 + −→ x 2 = (5, −3) = −→ x . Vˆa . y −→ x la` tˆo ’ ho . . p tuyˆe ´ n tı´nh cu ’ a hˆe . ( −→ x 1 , −→ x 2 ), hay −→ x biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . ( −→ x 1 , −→ x 2 ). Nhˆa . n xe´t. (1) Ca´ch biˆe ’ u diˆe ˜ n x = n  i=1 λ i x i no´i chung khˆong duy nhˆa ´ t. Vı´ du . . Trong khˆong gian vector thu . . c R 2 , xe´t 3 vector x 1 = (−1, 0), x 2 = (0, −1), x 3 = (1, 1). Khi d¯o´ vector khˆong 0 = (0, 0) biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . (x 1 , x 2 , x 3 ) b˘a ` ng ı´t nhˆa ´ t hai ca´ch sau: 0 = 0x 1 + 0x 2 + 0x 3 ; 0 = 1.x 1 + 1.x 2 + 1.x 3 . (2) Nˆe ´ u x = 0 ∈ V thı` v´o . i mo . i hˆe . vector (x i ) i=1,n ⊂ V , x bao gi`o . cu ˜ ng biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua (x i ) i=1,n . Vı´ du . . 0 = n  i=1 λ i x i , λ i = 0, ∀i = 1, n. Trong tru . `o . ng ho . . p na`y ta no´i 0 biˆe ’ u thi . tuyˆe ´ n tı´nh tˆa ` m thu . `o . ng qua hˆe . trˆen. Nˆe ´ u 0 co´ ı´t nhˆa ´ t hai ca´ch biˆe ’ u thi . tuyˆe ´ n tı´nh qua hˆe . (x i ) i=1,n thı` ta no´i 0 biˆe ’ u thi . tuyˆe ´ n tı´nh khˆong tˆa ` m thu . `o . ng qua hˆe . (x i ) i=1,n . Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 52 3. Khˆong gian vector 3.3.2 D - ˆo . c lˆa . p tuyˆe ´ n t´ınh v`a phu . thuˆo . c tuyˆe ´ n t´ınh. D - i . nh nghı ˜ a 3.4. Hˆe . n vector (n ≥ 1) (x i ) i=1,n trong K− khˆong gian vector V d¯u . o . . c go . i la` d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh nˆe ´ u vector khˆong chı ’ co´ duy nhˆa ´ t mˆo . t ca´ch biˆe ’ u thi . tuyˆe ´ n tı´nh qua hˆe . d¯o´ b˘a ` ng tˆo ’ ho . . p tuyˆe ´ n tı´nh tˆa ` m thu . `o . ng. Hˆe . khˆong d¯ˆo . c la . p tuyˆe ´ n tı´nh go . i la` hˆe . phu . thuˆo . c tuyˆe ´ n tı´nh. Nhu . vˆa . y, hˆe . (x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh khi va` chı ’ khi  n  i=1 λ i x i = 0 ∈ V  ⇒ (λ 1 = λ 2 = · · · = λ n = 0 ∈ K). Co`n hˆe . (x i ) i=1,n phu . thuˆo . c tuyˆe ´ n tı´nh nˆe ´ u va` chı ’ nˆe ´ u co´ ı´t nhˆa ´ t mˆo . t ho . vˆo hu . ´o . ng (λ i ) i=1,n khˆong d¯ˆo ` ng th`o . i b˘a ` ng khˆong sao cho n  i=1 λ i x i = 0 ∈ V . Vı´ du . . (1) Cho V = R 3 la` mˆo . t R− khˆong gian vector. Xe´t hˆe . {x 1 = (1, 1, 1), x 2 = (1, 1, 0), x 3 = (1, 0, 0)}. Gia ’ su . ’ tˆo ` n ta . i λ 1 , λ 2 , λ 3 ∈ R sao cho: λ 1 x 1 + λ 2 x 2 + λ 3 x 3 = 0 ⇔ (λ 1 + λ 2 + λ 3 , λ 1 + λ 2 , λ 1 ) = 0 ⇔      λ 1 + λ 2 + λ 3 = 0 λ 1 + λ 2 = 0 λ 1 = 0 ⇔      λ 1 = 0 λ 2 = 0 λ 3 = 0 Vˆa . y hˆe . d¯a ˜ cho d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh trong R 3 . (2) Cho V = R 2 la` mˆo . t R− khˆong gian vector. Xe´t hˆe . 3 vector : {x 1 = (1, −2), x 2 = (1, 4), x 3 = (3, 5)}. Gia ’ su . ’ co´ λ 1 , λ 2 , λ 3 ∈ R sao cho: λ 1 x 1 + λ 2 x 2 + λ 3 x 3 = 0 ⇔ (λ 1 + λ 2 + 3λ 3 , −2λ 1 + 4λ 2 + 5λ 3 ) = 0 ⇔  λ 1 + λ 2 + 3λ 3 = 0 −2λ 1 + 4λ 2 + 5λ 3 = 0 ⇔  λ 1 + λ 2 = −3λ 3 −2λ 1 + 4λ 2 = −5λ 3 ⇔      λ 1 = − 7 6 λ 3 λ 2 = − 11 6 λ 3 T`u . d¯ˆay ta co´ thˆe ’ cho . n ra rˆa ´ t nhiˆe ` u ho . vˆo hu . ´o . ng (λ i ) i=1,3 khˆong d¯ˆo ` ng th`o . i b˘a ` ng khˆong sao cho 3  i=1 λ i x i = 0 Vˆa . y hˆe . d¯a ˜ cho phu . thuˆo . c tuyˆe ´ n tı´nh. Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 3.3. Su . . phu . thuˆo . c tuyˆe ´ n t´ınh v`a d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. 53 Quy u . ´o . c. Hˆe . ∅ la` hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh. Vector 0 ∈ V la` tˆo ’ ho . . p tuyˆe ´ n tı´nh tˆa ` m thu . `o . ng cu ’ a hˆe . ∅ va` la` vector duy nhˆa ´ t biˆe ’ u thi . tuyˆe ´ n tı´nh qua hˆe . ∅. Nhˆa . n xe´t. (1) { −→ 0 } la` hˆe . phu . thuˆo . c tuyˆe ´ n tı´nh. (2) Nˆe ´ u hˆe . ( −→ x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh trong V thı` v´o . i mo . i −→ x ∈ V , −→ x co´ khˆong qua´ mˆo . t ca´ch biˆe ’ u thi . tuyˆe ´ n tı´nh qua hˆe . ( −→ x i ) i=1,n . (3) Cho hˆe . ( −→ x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh trong V va` −→ x ∈ V , nˆe ´ u −→ x biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . ( −→ x i ) i=1,n thı` ca´ch biˆe ’ u diˆe ˜ n d¯o´ la` duy nhˆa ´ t. Ch´u . ng minh. Gia ’ su . ’ −→ x biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . ( −→ x i ) i=1,n t´u . c la` tˆo ` n ta . i ca´c λ i ∈ K sao cho −→ x = λ 1 −→ x 1 + λ 2 −→ x 2 + · · · + λ n −→ x n . Nˆe ´ u ngoa`i ca´c λ i trˆen co`n tˆo ` n ta . i ca´c µ i ∈ K sao cho −→ x = µ 1 −→ x 1 + µ 2 −→ x 2 + · · · + µ n −→ x n . Thı` ta co´: λ 1 −→ x 1 + λ 2 −→ x 2 + · · · + λ n −→ x n = µ 1 −→ x 1 + µ 2 −→ x 2 + · · · + µ n −→ x n ⇔ (λ 1 − µ 1 ) −→ x 1 + (λ 2 − µ 2 ) −→ x 2 + · · · + (λ n − µ n )x n = −→ 0 ⇒            λ 1 − µ 1 = 0 λ 2 − µ 2 = 0 · · · λ n − µ n = 0 (do hˆe . ( −→ x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh) ⇔ λ i = µ i , ∀i = 1, n. Vˆa . y su . . biˆe ’ u thi . tuyˆe ´ n tı´nh cu ’ a −→ x qua hˆe . ( −→ x i ) i=1,n la` duy nhˆa ´ t. 3.3.3 V`ai t´ınh chˆa ´ t vˆe ` hˆe . phu . thuˆo . c tuyˆe ´ n t´ınh v`a hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. Tı´nh chˆa ´ t 3.7. (i) Hˆe . gˆo ` m mˆo . t vector { −→ x } d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh khi va` chı ’ khi −→ x = −→ 0 . (ii) Mo . i hˆe . vector ch´u . a −→ 0 d¯ˆe ` u phu . thuˆo . c tuyˆe ´ n tı´nh. Tı´nh chˆa ´ t na`y kha´ d¯o . n gia ’ n, ba . n d¯o . c tu . . ch´u . ng minh. Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 54 3. Khˆong gian vector Tı´nh chˆa ´ t 3.8. V´o . i hˆe . vector (x i ) i∈I tuy` y´ (I la` mˆo . t tˆa . p ho . . p bˆa ´ t ky` kha´c rˆo ˜ ng), hˆe . (x i ) i∈J go . i la` hˆe . con cu ’ a hˆe . (x i ) i∈I nˆe ´ u J ⊂ I. Khi d¯o´: (i) Nˆe ´ u hˆe . (x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh thı` mo . i hˆe . con cu ’ a no´ cu ˜ ng d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh. (ii) Nˆe ´ u co´ ı´t nhˆa ´ t mˆo . t hˆe . con phu . thuˆo . c tuyˆe ´ n tı´nh thı` hˆe . (x i ) i=1,n cu ˜ ng phu . thuˆo . c tuyˆe ´ n tı´nh. Ch´u . ng minh. Gia ’ su . ’ (x i ) i=1,n la` hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh va` (x j ) j∈J la` mˆo . t hˆe . con tuy` y´ cu ’ a no´, t´u . c la` J ⊂ I = {1, 2, ., n}. Ta cˆa ` n ch´u . ng to ’ (x j ) j∈J d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh. Thˆa . t vˆa . y, nˆe ´ u  j∈J λ j x j = 0 la` mˆo . t tˆo ’ ho . . p tuyˆe ´ n tı´nh b˘a ` ng 0 cu ’ a hˆe . (x j ) j∈J thı` 0 =  j∈J λ j x j +  i∈I\J 0.x i la` mˆo . t tˆo ’ ho . . p tuyˆe ´ n tı´nh b˘a ` ng 0 cu ’ a hˆe . (x i ) i=1,n . Ma` hˆe . (x i ) i=1,n la` hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh, suy ra λ j = 0, ∀j ∈ J, t´u . c la` (x j ) j∈J d¯ˆo . c lˆa . p tuyˆe ´ n. Vı` kha´i niˆe . m hˆe . phu . thuˆo . c tuyˆe ´ n tı´nh la` phu ’ d¯i . nh cu ’ a kha´i niˆe . m hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh nˆen hai kh˘a ’ ng d¯i . nh trong tı´nh chˆa ´ t na`y la` tu . o . ng d¯u . o . ng nhau. D - i . nh ly´ 3.2 (D - i . nh ly´ d¯˘a . c tru . ng cu ’ a hˆe . phu . thuˆo . c tuyˆe ´ n tı´nh). Hˆe . n vector (n ≥ 2) (x i ) i=1,n phu . thuˆo . c tuyˆe ´ n tı´nh khi va` chı ’ khi co´ (ı´t nhˆa ´ t) mˆo . t vector cu ’ a hˆe . biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua ca´c vector co`n la . i. Ch´u . ng minh. (⇒) Gia ’ su . ’ hˆe . (x i ) i=1,n phu . thuˆo . c tuyˆe ´ n tı´nh. Lu´c d¯o´ co´ ı´t nhˆa ´ t mˆo . t ho . vˆo hu . ´o . ng (λ i ) i=1,n khˆong d¯ˆo ` ng th`o . i triˆe . t tiˆeu sao cho 0 = n  i=1 λ i x i . Gia ’ su . ’ λ j = 0 ∈ K (1 ≤ j ≤ n). Khi d¯o´ n  i=1 λ i x i ⇒ −λ j x j =  i=j λ i x i ⇒ x j =  i=j  −λ i λ j  x i ; t´u . c la` x j biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . ca´c vector co`n la . i (x i ) i∈{1,2, .,n}\{j} . (⇐) Ngu . o . . c la . i, gia ’ su . ’ co´ mˆo . t vector cu ’ a hˆe . ch˘a ’ ng ha . n x j (1 ≤ j ≤ n), biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . ca´c vector co`n la . i, t´u . c la` co´ ca´c vˆo hu . ´o . ng λ i , i ∈ {1, 2, ., n}\{j} sao cho x j =  i=j λ i x i . Khi d¯o´ x j =  i=j λ i x i ⇒ 0 =  i=j λ i x i + (−1)x j D - ˆay la` mˆo . t tˆo ’ ho . . p tuyˆe ´ n tı´nh khˆong tˆa ` m thu . `o . ng b˘a ` ng 0 cu ’ a hˆe . (x i ) i=1,n . Vˆa . y hˆe . (x i ) i=1,n phu . thuˆo . c tuyˆe ´ n tı´nh. Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 3.4. Ha . ng cu ˙’ a mˆo . t hˆe . vector. 55 3.4 Ha . ng cu ˙’ a mˆo . t hˆe . vector. 3.4.1 Hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh tˆo ´ i d¯a . i. D - i . nh nghı ˜ a 3.5. Gia ’ su . ’ I la` mˆo . t tˆa . p ho . . p h˜u . u ha . n va` J ⊂ I. Cho hˆe . vector (x i ) i∈I tu`y y´ trong mˆo . t K− khˆong gian vector na`o d¯o´. Hˆe . (x j ) j∈J go . i la` hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a hˆe . d¯a ˜ cho nˆe ´ u no´ d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh va` nˆe ´ u thˆem bˆa ´ t ky` vector x i na`o, i ∈ I\J, va`o hˆe . con d¯o´ ta d¯ˆe ` u nhˆa . n d¯u . o . . c mˆo . t hˆe . phu . thuˆo . c tuyˆe ´ n tı´nh. Vı´ du . . Trong R 3 cho hˆe . 3 vector {x 1 = (1, 2, 3), x 2 = (2, 4, 6), x 3 = (3, 6, 9)}. Khi d¯o´ mˆo ˜ i hˆe . 1 vector {x 1 }, {x 2 }, {x 3 } d¯ˆe ` u la` hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh cu ’ a hˆe . d¯a ˜ cho. Ho . n n˜u . a, x 3 = 3x 1 , x 2 = 2x 1 , x 3 = 3 2 x 2 nˆen ca´c hˆe . con {x 1 , x 2 }, {x 1 , x 3 }, {x 2 , x 3 } d¯ˆe ` u phu . thuˆo . c tuyˆe ´ n tı´nh. Vˆa . y {x 1 }, {x 2 }, {x 3 } la` ca´c hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a hˆe . {x 1 , x 2 , x 3 } d¯a ˜ cho. Tı´nh chˆa ´ t 3.9. Nˆe ´ u hˆe . con (x i ) i=1,n cu ’ a hˆe . (x i ) i∈I ({1, 2, ., n} ⊂ I) la` mˆo . t hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i thı` mo . i vector x i , i ∈ I d¯ˆe ` u biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . con d¯o´ va` ca´ch biˆe ’ u thi . la` duy nhˆa ´ t. Tı´nh chˆa ´ t na`y la` hˆe . qua ’ tru . . c tiˆe ´ p cu ’ a D - i . nh nghı ˜ a 3.5 va` D - i . nh ly´ 3.2. Bˆo ’ d¯ˆe ` 3.1 (Bˆo ’ d¯ˆe ` co . ba ’ n vˆe ` su . . phu . thuˆo . c tuyˆe ´ n tı´nh). Cho (x 1 , x 2 , ., x m ) va` (y 1 , y 2 , ., y n ) la` hai hˆe . vector trong khˆong gian vector V . Gia ’ su . ’ hˆe . (x i ) i=1,m d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh va` mˆo ˜ i x i (i = 1, m) d¯ˆe ` u biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua hˆe . (y j ) j=1,n . Khi d¯o´ m ≤ n. D - i . nh ly´ 3.3. Mo . i hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a mˆo . t hˆe . h˜u . u ha . n vector trong mˆo . t K− khˆong gian vector tu`y y´ d¯ˆe ` u co´ sˆo ´ vector b˘a ` ng nhau. Ch´u . ng minh. Gia ’ su . ’ (x i ) i∈I la` mˆo . t hˆe . vector h˜u . u ha . n. Nˆe ´ u x i = 0 v´o . i mo . i i ∈ I thı` (x i ) i∈I chı ’ co´ mˆo . t hˆe . d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i duy nhˆa ´ t la` ∅ va` kh˘a ’ ng d¯i . nh cu ’ a d¯i . nh ly´ la` hiˆe ’ n nhiˆen. Gia ’ su . ’ hˆe . (x i ) i∈I co´ ch´u . a vector kha´c khˆong. Khi d¯o´ ca´c hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a (x i ) i∈I co´ ı´t nhˆa ´ t mˆo . t vector. Gia ’ su . ’ (x j ) j∈J 1 va` (x j ) j∈J 2 la` hai hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a (x i ) i∈I (J 1 ⊂ I, J 2 ⊂ I) v´o . i sˆo ´ vector lˆa ` n lu . o . . t la` m va` n (m, n ≥ 1). Vı` (x j ) j∈J 2 d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i nˆen mo . i x j , j ∈ J 1 d¯ˆe ` u biˆe ’ u thi . tuyˆe ´ n tı´nh d¯u . o . . c qua (x j ) j∈J 2 . Ma` (x j ) j∈J 1 d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh, do d¯o´ theo Bˆo ’ d¯ˆe ` 3.1, ta co´ m ≤ n. Tu . o . ng tu . . cu ˜ ng co´ n ≤ m. Vˆa . y n = m. Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh 56 3. Khˆong gian vector 3.4.2 Ha . ng cu ˙’ a mˆo . t hˆe . vector. D - i . nh nghı ˜ a 3.6. Cho V la` mˆo . t K− khˆong gian vector, (x i ) i∈I la` mˆo . t hˆe . vector bˆa ´ t ky` trong V . Nˆe ´ u hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a (x i ) i∈I co´ sˆo ´ phˆa ` n tu . ’ h˜u . u ha . n b˘a ` ng r thı` r d¯u . o . . c go . i la` ha . ng cu ’ a hˆe . (x i ) i∈I . Kı´ hiˆe . u: rank((x i ) i∈I ) = r. Vı´ du . . Xe´t la . i hˆe . vector {x 1 = (1, 2, 3), x 2 = (2, 4, 6), x 3 = (3, 6, 9)} cu ’ a R 3 . Vı` {x 1 } la` mˆo . t hˆe . con d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh tˆo ´ i d¯a . i cu ’ a hˆe . {x 1 , x 2 , x 3 } nˆen rank(x 1 , x 2 , x 3 ) = 1. Nhˆa . n xe´t. Khi cho (s) = (x i ) i=1,n la` mˆo . t hˆe . vector trong V va` r =rank(s) thı`: (i) r ≤ n, (ii) Nˆe ´ u (s) = (x i ) i=1,n d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh thı` rank(s) = r = n. 3.4.3 C´ac hˆe . vector trong K n . Trong khˆong gian K n xe´t m vector sau: a 1 = (a 11 , a12, ., a 1n ) a 2 = (a 21 , a22, ., a 2n ) . a m = (a m1 , am2, ., a mn ) Go . i A = (a ij ) m×n la` ma trˆa . n cˆa ´ p m × n trˆen K ma` ca´c do`ng chı´nh la` a 1 , a 2 , ., a m . Khi d¯o´ ta co´ ca´c kh˘a ’ ng d¯i . nh sau d¯ˆay: D - i . nh ly´ 3.4. V´o . i hˆe . (a 1 , a 2 , ., a m ) va` ma trˆa . n A d¯u . o . . c d¯i . nh nghı ˜ a nhu . trˆen, ta co´: (1) Hˆe . (a 1 , a 2 , ., a m ) d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh trong K n ⇔ rank(A) = m. (2) Hˆe . (a 1 , a 2 , ., a m ) phu . thuˆo . c tuyˆe ´ n tı´nh trong K n ⇔ rank(A) = m. D - i . nh ly´ 3.5. Ha . ng cu ’ a mˆo . t ma trˆa . n cˆa ´ p m × n trˆen K b˘a ` ng ha . ng cu ’ a hˆe . vector cˆo . t (tu . o . ng ´u . ng, do`ng) cu ’ a no´ trong K m (tu . o . ng ´u . ng, K n ). T`u . 2 d¯i . nh ly´ trˆen ta suy ra mˆo . t ca´ch d¯ˆe ’ xe´t tı´nh d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh hay phu . thuˆo . c tuyˆe ´ n tı´nh cu ˜ ng nhu . tı`m ha . ng cu ’ a mˆo . t hˆe . vector trong K n la` d¯i tı`m ha . ng cu ’ a ma trˆa . n d¯u . o . . c ta . o nˆen bo . ’ i ca´c vector d¯o´. Vı´ du . . Xe´t tı´nh d¯ˆo . c lˆa . p tuyˆe ´ n tı´nh hay phu . thuˆo . c tuyˆe ´ n tı´nh va` tı`m ha . ng cu ’ a ca´c hˆe . vector sau: Ba`i gia ’ ng D - a . i sˆo ´ tuyˆe ´ n tı´nh [...]... = 2 < 3 (sˆ vector cua hˆ), do d´ (v1, v2 , v3 ) phu thuˆc a o ¯o 4 ´ tuyˆ n tı trong R va rank(v1, v2 , v3 ) = 2 e ´nh ` ´ ` ˙ ’ Co so - Sˆ chiˆu - Toa d o cua khˆng gian vector o e o ¯ˆ ˙ ’ 3.5 3.5.1 ˙ ˙ ’ ’ Co so cu a khˆng gian vector o - ˜ Dinh nghı a 3.7 Cho K− khˆng gian vector V Hˆ vector = (e1 , e2 , , en ) o e o.c goi la mˆt co so cua V nˆ u d oc lˆp tuyˆ n tı va moi vector ´ ´ ’... khˆng gian vector, (S) la mˆt hˆ vector ` o o ` o e o.c goi la hˆ sinh cua khˆng gian vector V nˆ u moi x ∈ V , x ´ ’ trong V (S) d ` e ¯u o e cu ng biˆ u thi tuyˆ n tı d o.c qua hˆ (S) d´ ’ ˜ ´ ´nh ¯u bao gi` o e e e ¯o Nhˆn xe t a ´ ’ ’ (1) Mˆt co so cua K− khˆng gian vector V la mˆt hˆ sinh nhu.ng d ` u ngu.o.c o o ` o e ¯iˆ e lai khˆng d ´ ng o ¯u ’ ’ (2) Trong K− khˆng gian vector. .. K3 nhˆn hˆ vector gˆm 3 vector : o a e o → → − e2 e3 (e) = {→ = (1, 0, 0), − = (0, 1, 0), − = (0, 0, 1)} e1 -ˆ ` o ` ’ e lam co so nˆn dimV = 3 Day la mˆt khˆng gian h˜.u han chiˆu ` o u e 2 3 (2) K− khˆng gian vector K[x] nhˆn hˆ vector (e) = {1, x, x , x , } la mˆt o a e ` o so Day la mˆt khˆng gian vˆ han chiˆu ` ’ -ˆ ` o o o e co - ` Dinh ly 3.7 Cho V la mˆt K− khˆng gian vector n chiˆu... khˆng gian n chiˆu (n ∈ N tuy ´ ) d o.c goi chung la ca c a ´ o e ` y ¯u ` ´ u han chiˆu ` khˆng gian h˜ o u e ´ ´ ’ng - o Bai gia Dai sˆ tuyˆ n tı ` e ´nh 60 3 Khˆng gian vector o ` ´ e ´ (2) V goi la khˆng gian vˆ han chiˆu, kı hiˆu dim V = ∞, nˆ u no khˆng h˜.u o e e ´ o u ` o ` ` ’ ’ han chiˆu, t´.c hˆ vector co so cua V co vˆ han phˆn tu e u e ´ o a ’ Vı du ´ ` (1) K− khˆng gian vector. .. o ’ (3) Trong K− khˆng gian vector Kn , hˆ vector o e (e) = {e1 = (1, 0, 0, , 0, 0), e2 = (0, 1, 0, , 0, 0), , en = (0, 0, 0, , 0, 1)} ´ ’ ` ` ¯u ` ’ ’ la mˆt co so va co so nay con d o.c goi la co so chuˆ’n t˘ c cua Kn ` o ’ ’ ` a a ’ (4) Trong K− khˆng gian vector K[x], hˆ vector (e) = {1, x, x2 , x3 , } la mˆt o e ` o so ’ co 3.5.2 ˙ ’ Hˆ sinh cu a mˆt khˆng gian vector e o o - ˜ Dinh... cua V d` u biˆ u thi tuyˆ n tı qua ¯ˆ e e’ e ´nh Nhˆn xe t a ´ ´ ` ’ ’ (1) V´.i mˆt khˆng gian vector bˆ t ky bao gi` cu ng tˆn tai mˆt co so cua o o o a ` o ˜ o o no ´ ´ ’ ’ (2) Co so cua mˆt khˆng gian vector la khˆng duy nhˆ t o o ` o a Vai vı du ` ´ (1) Trong K− khˆng gian vector K3 cho hˆ gˆm 3 vector : o e ` o − → → (e) = {→ = (1, 0, 0), − = (0, 1, 0), − = (0, 0, 1)} e e e 1 2 3 ˜ a ´... Khˆng gian h˜.u han v` vˆ han chiˆu o e o u a o e - ˜ Dinh nghı a 3.9 Cho V la mˆt K− khˆng gian vector va n la mˆt sˆ tuy ´ ` o o ` ` o o ` y ´ ` ´ e ’ ’ ’ (1) Ta bao V la mˆt khˆng gian n chiˆu (n ≥ 1) nˆ u hˆ vector co so cua V ` o o e e b˘ ng n Ta cu ng bao sˆ chiˆu cua V la n va kı hiˆu dim ˜ ´ a ’ ` ´ ` ’ o co sˆ phˆn tu a ´ o ` e ’ ` ` ´ e V = n ´ ’ ` Khˆng gian khˆng (chı gˆm mˆt vector. .. a o ´ ng minh r˘ ng R cung v´.i hai phe p toa n trˆn la mˆt khˆng gian ` Ch´ u a o ´ ´ e ` o o + ` c vector thu 3.2 Xe t xem R2 cung v´.i hai phe p toa n sau co lˆp thanh mˆt R− khˆng gian ´ ` o ´ ´ ´ a ` o o vector khˆng? o ´ ´ ’ng - o Bai gia Dai sˆ tuyˆ n tı ` e ´nh ´ ` ˙ ’ o e o 3.5 Co so - Sˆ chiˆu - Toa d o cua khˆng gian vector ¯ˆ ˙ ’ 63 a (a, b) + (c, d) = (a, b); λ(a, b) = (λa, λb),... → + x → + x → − − − − − ∀x ´ x 1 2 3 1 e1 2 e2 3 e3 ’ ’ Vˆy (e) la mˆt co so cua K3 a ` o (2) Trong K− khˆng gian vector K3 cho hˆ gˆm 3 vector : o e ` o − − → (u) = {→ = (1, 1, 1), → = (1, 1, 0), − = (1, 0, 0)} u u u 1 ´ ´ ’ng - o Bai gia Dai sˆ tuyˆ n tı ` e ´nh 2 3 58 3 Khˆng gian vector o → − − − Ta co λ1 → + λ2 → + λ3 → = 0 ´ − u1 u2 u3   λ1 + λ 2 + λ 3 = 0  λ1 = 0   → − ⇔ (λ1 + λ2 +... nghı a 3.11 Cho V la mˆt K− khˆng gian vector n chiˆu, (e) = ` o o e → → − − − → − − → ` ’ ’ a e’ {→, →, , − } va (e ) = { e1 , e2 , , en } la hai co so cua V Ma trˆn chuyˆ n e1 e2 en ` so t` (e) sang (e ) la mˆt ma trˆn vuˆng cˆ p n v´.i cˆt th´ j la toa d o cua ´ co ’ u ` o a o a o o u ` ¯ˆ ’ → − ´ ’ vector ej d o i v´.i co so (e) ¯ˆ o Vı du Trong K− khˆng gian vector K3 : ´ o − → → (e) = {→ . cu ˙’ a khˆong gian vector. 3.5.1 Co . so . ˙’ cu ˙’ a khˆong gian vector. D - i . nh nghı ˜ a 3.7. Cho K− khˆong gian vector V . Hˆe . vector  = (e 1. Khˆong gian vector con. D - i . nh nghı ˜ a 3.2. Mˆo . t tˆa . p ho . . p con W = ∅ cu ’ a K− khˆong gian vector V d¯u . o . . c go . i la` khˆong gian vector